Modular quasi-Hopf algebras and groups with one involution

Abstract: In a previous paper the authors constructed a class of quasi-Hopf algebras $D^{\omega}(G, A)$ associated to a finite group $G$, generalizing the twisted quantum double construction. We gave necessary and sufficient conditions, cohomological in nature, that the corresponding module category $Rep(D^{\omega}(G, A))$ is a modular tensor category.\ In the present paper we verify the cohomological conditions for the class of groups $G$ which \emph{contain a unique involution}, and in this way we obtain an explicit construction of a new class of modular quasi-Hopf algebras.\ We develop the basic theory for general finite groups $G$, and also a parallel theory concerned with the question of when $Rep(D^{\omega}(G, A))$ is super-modular rather than modular. We give some explicit examples involving binary polyhedral groups and some sporadic simple groups.